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For example, we have the bridge between elliptical curves and modular forms, and we have the Langlands program. This is a famous example. What significant tie-ins are there between your field and others?

Perhaps this isn't proper /r/puremathematics content, but, as an early graduate student that is forced to deal with each course individually (analysis, topology, algebra, &c) as being on its own by virtue of the way courses are structured, I find this to be an interesting question and hope that it's worthy of more general discussion.

When I was an undergraduate, I took my school's graduate level point-set topology class, in which our professor have us an example of a result in number theory that was several dozen pages from that point of view but was perhaps two pages in a topological framework, and I thought that that was fascinating. (I wish I had saved it, but I've lost the handout and he never gave out PDFs.) This isn't a good example of an answer to my question since it's more of a coincidence than a "bridge", but I'd be interested to hear examples others may have in mind.

The nullstellensatz is the link between geometry and (commutative) algebra. Over an algebraically closed field it gives (basically) a bijection between points on a geometric object and maximal ideals in a certain ring. It also will help convince you that the basic definitions of algebraic geometry mean something.

We don't use Spur in English -- it's called a "trace." On the other hand, people do actually use the word Nullstellensatz and less occasionally speak of the Hauptidealsatz or Kronecker's Jugendtraum, or use an ansatz to solve a problem. Also the blackboard bold "Z" by which we denote the integers comes from the word Zahlen, but if you speak German you probably already knew that.

I'm taking a course on Grobner bases, and we use the Nullstellensatz critically. I pronounce it "null-STELL-in-sats" as it reads in English, but this crazy European dude in my class pronounces it as "NOOLH-schtell-ahn-satz" and I lol every time. He's probably right though.

Nullstellensatz also gives a link between Model theory and algebraic geometry. There is a pretty cool proof of the weak Nullstellensatz using a model theoretic property of algebraically closed fields known as Model Completeness.

Some of my favorites have already been mentioned, but I'll add a few more.

Iwasawa Theory and the Bloch-Kato Conjecture - Both of these relate L-functions (inherently ANALYTIC objects) to the sizes of Selmer groups (inherently ALGEBRAIC objects).

Gauss-Bonnet Theorem: This tells you that if you take the curvature of a surface and integrate over the entire surface, then you get 2 pi times the Euler characteristic. This can be viewed as a bridge between the local geometry of a surface and the global topology of a surface.

Hirzebruch Signature Theorem - This theorem involves two topological invariants of a manifold, one of which comes purely from algebraic topology and one of which is related to algebraic topology but also involves the Bernoulli numbers (which are typically considered to be number-theoretic in nature).

The Proof of the Classification of Complex Semisimple Lie Algebras - The proof begins with Lie algebras, turns into a problem about certain highly symmetric configurations of vectors in Euclidean space (called root systems), and is concluded via solving a purely combinatorial problem (classifying Dynkin diagrams).

Here are some examples that a low-dimensional topologist might be familiar with in addition to the already-mentioned Atiyah-Singer index theorem:

Gauge theory: one of the best ways to construct algebraic invariants of low-dimensional manifolds is to study solutions of elliptic PDEs on them (the Yang-Mills equation, the Seiberg-Witten equations, pseudoholomorphic curves, etc.); this often associates Floer homology groups to 3-manifolds and group homomorphisms to cobordisms between them.

Knot homologies: powerful knot invariants such as Khovanov homology can be constructed from representations of quantum groups. The proof that Khovanov homology is an "unknot detector" also relies heavily on gauge theory.

Geometric group theory: a hyperbolic 3-manifold is determined up to isometry by its fundamental group, and so you can understand its geometry by group-theoretic techniques as in the recent proof of the virtual Haken conjecture.

De Rham's Theorem gives an explicit link between differential topology/geometry and topology.

What's interesting about this is first of all how the connection is done (Stoke's Theorem in full generality), as well as providing a link between how to view a topological space with and without a differentiable structure.

This gives a very handy tool in understanding a topological space with a differentiable structure better, as you can attack it with either of the two theories and get results for both.